Method for determining the damped natural frequencies of a dynamic system

ABSTRACT

The present invention is a novel device, system, and method for determining the damped natural frequencies of a dynamic system whose characteristics are available in the form of a “frequency response function” FRF. According to an exemplary embodiment of the present invention, the method identifies the dampened natural frequencies associated with the poles and zeros of a transfer function. The method is especially useful for analysis of measurements that contain some degree of contamination due to noise or non-linear effects. It is based on a set of rules that may be more successful than a direct approach based on a least squares criteria.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims priority to U.S. Patent Application No.60/801,930 filed May 19, 2006 entitled “Method for Determining theDamped natural Frequencies of a Dynamic System”, which is incorporatedfully herein by reference.

TECHNICAL FIELD

The present invention relates to analyzing a frequency response functionand more particularly, to a device, method, and system for analyzing afrequency response function for a dynamic system to determine the dampedfrequencies.

BACKGROUND INFORMATION

A frequency response function is one form of a description for thedynamics of a linear system. The dynamic relationship between twovariables associated with such a system as an analytical expression isknown as a transfer function. The transfer function is one way ofrepresenting the differential equations that describe a relationshipbetween state variables or between a system input and a state variable.A linear system's transfer function can be represented as a ratio ofpolynomials in the complex Laplace variable s. In the field of modalanalysis a technique for deriving a transfer function from an FRF istermed a “condensation algorithm.” Many known condensation algorithmsattempt to find the optimal set of coefficients for the s polynomialusing criteria such as least squares. Attempts to use existing knowntechniques in situations where the number of available measurementlocations is limited have yielded unsatisfactory results. When the goalof an analysis is to automate commissioning of controllers or to quicklydetermine a basic model for system dynamics, a method that develops auseful result from a small number of measurements may be desirable. Insome applications the ability to identify a finite number of frequencieswhere the transfer function dynamics have lightly damped resonances andanti-resonances is more important than the ability to identify exactlythe damping factors associated with these resonances/anti-resonances.Such is the case when automating the process of tuning control loops inmotion control systems. In auto-tuning applications, identification ofall the significant lightly damped resonant frequencies is arequirement. The term significant indicates a resonance that has someimpact on the stability or performance of a closed loop system. It isalso anticipated that a multiple step approach where the real andimaginary components of each pole are identified in separate steps maylead to an overall better result. In this case heuristic rules foridentifying the damped natural frequencies of poles and zeros correspondto determination of the imaginary components of the roots. Next ageneral constrained optimization can be performed to identify the realcomponents by minimizing the least squared error between the originalFRF and one derived for the selection of real components.

There exist a number of methods for identifying the dynamics of anoscillatory mechanical system by analyzing an FRF. Among publicationsthat document these methods are those by Drs. Randy Allemang and DavidBrown. These methods either operate directly on the frequency domain FRFor operate on the impulse response time domain function obtained byapplying the inverse Fourier transform to the FRF. These methods useleast squares criteria to fit a large number of measurements fromvarious measurement points on the mechanical system to determine thesystem eigenvalues. In a second step the residues are determined fromwhich the transfer function matrix is derived. The same methods appliedto a limited number of input-output locations have failed to provide theneeded results.

In U.S. Pat. No. 6,347,255 to Moser, Moser describes a technique foridentifying “poles” and “zeroes” from an FRF measurement. Moser'sapproach is heuristic and is designed specifically for the case of servotuning. Software that implements Moser's approach occasionally fails toidentify a pole or zero whose presence affects the tuning of the servocontroller.

The current invention may also uses a heuristic approach but providesadditional action and aspects to Moser's approach by imposing morerestrictions on the conditions for identifying a pole/zero to gainimproved immunity to noise. The result may provide a more robust methodthat can detect more actual poles while being less susceptible to falseresults, i.e. identification of a pole that is not a system pole butrather a consequence of noise in the measurement or a manifestation ofnon-linear effect in the system.

Accordingly, a need exists for a device, method, and system for quicklyand efficiently determining the damped frequencies of a dynamic system.There may be an additional need to prevent misidentifying or identifyingincorrect damped frequencies.

SUMMARY

The present invention is a novel device, system, and method fordetermining the damped natural frequencies of a dynamic system.According to an exemplary embodiment of the present invention, themethod may produce a frequency response function for the dynamic systemand smooth the frequency response function. The method identifies a setof possible candidate frequency values by determining when a derivativeof the frequency response amplitude function changes sign, that is theamplitude derivative function traverses zero. The frequencies are sortedinto two lists: a list of “poles” where the derivative functiontraverses zero with negative slope, and “zeroes” where the derivativefunction traverses zero with positive slope. The method may sort the setof possible candidate frequency values from highest to lowest amplitude,in the case of “poles” and lowest to highest amplitude in the case of“zeroes”. The method may eliminate the least suitable candidates forpoles and zeroes using similar and analogous techniques. For the sake ofclarity this description describes the case for poles, while completelyanalogous approaches may be used for elimination of least suitablecandidates for zeroes. The method may identify the next candidate in thesorted candidate frequency values starting from the beginning of the setthat has not been eliminated. The method determines a range of influenceby identifying a lower root frequency and higher root frequency. Themethod eliminates the candidate frequency value by comparing thecandidate frequency value with a maximum amplitude in the range ofinfluence and if the candidate frequency value is eliminated returns tothe action of verifying the next candidate. The method may check for adirection of a change of phase in the range of influence thatcorresponds to the expected behavior for a pole or zero: typicallypositive change in phase for zeros and negative change in phase forpoles. When direction is opposite that expected, the method mayeliminate the candidate frequency value and the return to the action ofverifying the next candidate.

The method may determine which frequency candidate values of lessersignificance are in the range of influence by identifying frequencycandidate values occurring later in the list of frequency candidatevalues. When the frequency candidate value occur in the range ofinfluence of the currently considered candidate, the method mayeliminate the later candidate frequency value and then return to theaction of identifying the next candidate. The method may determine ifthere exist frequency candidate values that are of greater significancewithin the range of influence of the currently considered pole/zero byidentifying frequency candidate values occurring earlier in the list offrequency candidate values. When an earlier value is not designated ashaving been eliminated and is within the range of influence of thecurrently considered candidate, the method may eliminate the currentlyconsidered candidate frequency value and return to the action ofidentifying the next candidate. The method may determine if thefrequency is in the range of influence of the frequency responsefunction prior to the action of smoothing. The method may then eliminatethe frequency candidate value by comparing a noise level in thefrequency response function to the amplitude in the region of acandidate pole and when the candidate frequency value is eliminatedreturning to the action of identify the next candidate. The methodreturns to the action of identifying the next candidate until allfrequency candidate values have been identified.

It is important to note that the present invention is not intended to belimited to a system or method which must satisfy one or more of anystated objects or features of the invention. It is also important tonote that the present invention is not limited to the exemplaryembodiments described herein. Modifications and substitutions by one ofordinary skill in the art are considered to be within the scope of thepresent invention, which is not to be limited except by the followingclaims.

BRIEF DESCRIPTION OF THE DRAWINGS

These and other features and advantages of the present invention will bebetter understood by reading the following detailed description, takentogether with the drawings herein:

FIG. 1 is a diagram of an exemplary candidate pole that passes theparabola fit test according to an embodiment of the invention.

FIG. 2 is a diagram of an exemplary candidate zero that fails theparabola fit test according to an embodiment of the invention.

FIG. 3 is a diagram of an exemplary zero that passes the parabola fittest according to an embodiment of the invention.

FIGS. 4A and 4B are a flowchart of a method for determining the dampednatural frequencies of a dynamic system according to an exemplaryembodiment of the present invention.

DETAILED DESCRIPTION

The present invention is a novel device, system, and method fordetermining the damped natural frequencies of a dynamic system whosecharacteristics are available in the form of a “frequency responsefunction” FRF. According to an exemplary embodiment of the presentinvention, the method identifies the dampened natural frequenciesassociated with the poles and zeros of a transfer function. The methodis especially useful for analysis of measurements that contain somedegree of contamination due to noise or non-linear effects. It is basedon a set of rules that may be more successful than a direct approachbased on a least squares criteria.

Pre Conditions

The algorithm for identifying the damped natural frequencies associatedwith the poles and zeroes of an oscillatory mechanical system is basedon some observations about mechanical systems. For the case of a systemthat relates a force or torque input to a position or speed output, thefollowing rules may apply.

-   -   The sequence of the appearance of poles and zeros with        increasing frequency is not fixed and depends on the specific        characteristics of the dynamics system: e.g. a sequence of        z-p-z-p is possible, but so is a sequence z-p-p-z etc.    -   The amplitude of the FRF at a damped natural frequency reaches        an extreme value.    -   The amplitude experiences a local maximum at the damped natural        frequency of a pole.    -   The amplitude experience local minimum at the damped natural        frequency of a zero.    -   The phase of the FRF decreases in the neighborhood of a pole.    -   The phase of the FRF increases in the neighborhood of a zero        (for most systems).    -   Measurements often contain “noise” that can be falsely deemed a        pole or zero.    -   Noise effects are reduced by smoothing the FRF—smoothing is some        type of averaging of the value at a given frequency with values        at neighboring frequencies.    -   Too much smoothing distorts the shape of the FRF.    -   Smoothing amplitude and phase results in a different sort of        distortion than the type of distortion accompanying smoothing of        the complex values.        frequency, and to indicate whether or not it is eliminated as a        candidate.        Determination of Damped Natural Frequency

The algorithm for identifying damped natural frequencies of poles andzeroes may use nearly identical rules for finding zeros and poles. Asingle algorithm that handles both types of roots may be implementedwhere a change in parameterization configures the algorithm to thespecific task of either zero or pole finding.

A summary of the algorithm 400 follows. The summary will pertain topoles; modifications to account for zeros are indicated in squarebrackets. The basic idea of the algorithm is to identify a list ofcandidate poles/zeroes and then eliminate candidates from the list byimposing a series of stricter and stricter criteria. The candidates maybe represented as arrays of objects where each object has member data toindicate the frequency, and to indicate whether or not it is eliminatedas a candidate.

1. Obtain an FRF H (block 402).

2. Smooth H moderately to obtain H_(s)(block 404). This is done toremove fluctuations and noise from the FRF, while maintaining theoverall shape. In one method, the smoothing uses an averaging windowwhose width increases linearly with frequency. In this embodimentsmoothing is performed by transforming the complex valued FRF to polarcoordinates, smoothing amplitude and phase separately and re-combiningto form a complex value. In some cases the smoothing operation may alsobe used to smooth the complex values

Smoothing is implemented as a method of a software class forrepresenting a FRF. Various types of smoothing are made available toaddress a number of different requirements. In general smoothingconsists of performing a windowed average over a range of frequencies todetermine the smoothed value at the center frequency. The width of thesmoothing Window in terms of frequency range may be fixed or may vary asa function of frequency. Linear and log-wise functions of window widthas a function of frequency are some examples of how the window width maybe varied. Such variable width windows are especially useful with anFRF, because the typical width of a resonance or anti resonancegenerally increases with frequency. This means that too much smoothingat a low frequency will completely distort the shape of the resonanceand anti-resonances at the low frequencies. Similarly more smoothing isrequired at high frequencies to effectively eliminate the fluctuationsthat can occur due to imperfections in the measurement, Additionalflexibility in smoothing allows the user to specify the shape of thesmoothing window W as a function of the local frequency f measured fromthe window start frequency, e.g. flat: W(f)=const, triangularW(f)=const+a*f (fmin<f<fctr), W(f)=const+a*f (ctr)−a*(f−fmin)(fctr<f<fmax), or exponential: W(f) const+a exp(−(fctr−f)|*b).Additionally, the smoothing may be applied to the complex valued FRF, ormay be applied to the amplitude and phase functions separately and thenrecombined to form a complex number.

3. Identify candidate values by identifying locations where thederivative of the amplitude |H_(s)| are zero and where the slope of thederivative is negative [positive for zeros], Use linear interpolationfor cases where the derivative of the sequence traverses zero (block406). A method of the FRF class is especially useful for locatingfrequencies where some aspect of the FRF (real, imaginary, amplitude,phase, dB-amplitude) traverses a certain value.

4. Sort the frequencies in order from highest to lowest amplitude [forzeros: lowest to highest amplitude] and store sorted result in an arrayof candidate objects C. C.ω=root frequency, C.ok=logical variable toindicate whether candidate has been eliminated: C.ok initializes to TRUE(block 408).

5. Initialize the active root as the first in the sorted list Cn=C[0](block 410).

6. Find the next candidate in the list that has not yet been eliminated(block 412).

7. Determine the “range of influence” of the root Cn. The range ofinfluence is represented as a pair of frequencies: one lower than theroot frequency: ω_(left) and one higher than the root frequency:ω_(right), that approximate the neighborhood of the root. The lowerfrequency ω_(left) is determined by finding the highest frequency belowCn.ω where ∥H_(s)(ω)∥ passes through a factor k times the amplitude atthe candidate frequency k*∥Hs(Cn.ω)∥ with a positive slope [negativeslope for zeros]. The upper frequency ω_(right) is determined by findingthe lowest frequency above Cn.ω where ∥H_(s)(ω)∥ passes throughk*∥H_(s)(Cn.ω)∥ with a negative slope [positive slope for zeros]. Forepoles, the factor k is a number between 0 and 1, the value 0.7 hasproven to work well (block 414). For zeroes, the factor k is a numbergreater than 1, the value 1.4 has proven to work well.

8. Designate Cn as eliminated (Cn.ok=FALSE) if the range of influence istoo large relative to the root frequency. This accounts for cases wherethe candidate had been caused by a very small deviation along a moregeneral trend (upward or downward) where the size of the deviation afterfiltering is less than the factor employed in the previous step (e.g.0.7 times peak value causes the algorithm to find a location on thelarger trend) (block 416). While such cases may indicate an actualpole/zero, they are eliminated as their amplitude effect isinsignificant relative to the larger trend.

9. Designate Cn as eliminated if the amplitude at the candidatefrequency is not nearly as large as the maximum amplitude in the rangeof influence. This would mean that the candidate is not significantrelative to noise. If the candidate Cn is eliminated at this step, thenreturn to Step 6, otherwise continue to step 10 (block 418).

10. Check that the direction of the change of phase in the range ofinfluence is in the correct direction (positive for poles, negative forzeros). The net change in phase is determined by fitting a straight lineand intercept to the phase in the region of influence. This is doneusing a linear least squares solution using well known techniques. Priorto fitting a line to the phase response it must undergo an “unwrap”operation to avoid step discontinuities for cases where the phasetraverse 180 degrees. Unwrapping consists of locating instantaneousphase changes that suggest a traversal through 180 degrees. The generalphase trend is the slope of the straight line. If the slope is in thewrong direction (has the incorrect sign), then the candidate iseliminated and the processing returns to Step 6 (block 420).

11. Check to see if any candidates of lesser significance are in therange of influence. Eliminate less significant poles [zeros] whosefrequency is in the range of influence of Cn. A less significant pole[zero] is identified as one that occurs later in the list of candidates.This means a candidate whose amplitude is lower [higher] than theamplitude of Cn (block 422).

12. Check if any candidates that have not been eliminated of greatersignificance are in the range (block 424). This is accomplished bysearching the list of candidates from the beginning of the list to thecandidate nearest Cn and testing if any of these are in the range of Cn.If a candidate of greater significance is found, then eliminate Cn andreturn to Step 6. Such a situation can occur even though Step 11 willhave been performed on all previous candidates. This is the case whenthe range of influence of the lower amplitude candidate is wider thanthe range of influence of the higher amplitude candidate.

13. Eliminate any frequency shifting that might have been caused by thesmoothing in step 2 (block 426). The original un-smoothed FRF isexamined to find the frequency in the range of influence where theamplitude of the original FRF H experiences a maximum [minimum]. Changethe frequency of Cn.ω to be this new frequency and re-formulate therange of influence by shifting the range to be centered on the adjustedfrequency while maintaining its width.

14. Eliminate the candidate if the noise level in the original signal issignificant relative to the overall amplitude trends in the region ofthe pole (block 428). This is accomplished by fitting two parabolas tothe amplitude response. The parabolas are formulated as a function ofdistance from the window start frequency with three parameters:A(Δf)=c₀+c₁*Δf+c₂*Δf². The first parabola spans the frequency range fromthe range of influence lower limit to the frequency just below that ofCn.ω. The second parabola spans the frequency range from the frequencyjust above that of Cn.ω to the range of influence upper limit. Thus thefrequency of the pole/zero is not included in the parabola fit. Theparabola is intended to identify whether there is a trend at adjacentfrequencies leading to the pole/zero. If the standard deviation of theoriginal signal ∥H∥ measured over the frequency range of influence isgreater than twice the standard deviation of the two parabolas, it maybe interpreted to mean that the amplitude of the candidate root issimilarly significant relative to the noise and candidate Cn iseliminated. If the orientation of the parabolas does not represent apole [zero] then the candidate is eliminated. The orientation of theparabolas requires that the value of each the parabola at the locationadjacent to the root frequency be the greater [less] than the value ofeach parabola at the location farthest from the root frequency. Anexample of a candidate pole that passes the parabola test is shown inFIG. 1. An example of a candidate zero that fails the parabola test isshown in FIG. 2. An example of a zero that passes the parabola fit testis shown in FIG. 3.

15. If there are remaining candidates to evaluate, return to step 6,otherwise continue to step 16 (block 430).

16. Consolidate all candidates that had not been eliminated. These arethe poles. Repeat steps 3 through 15 for the zeros (block 432).

Determination of Real Components of Roots

In some applications, such as automated servo tuning of speed andposition controllers, identification of the damped natural frequency ofpoles and zeros is sufficient information to proceed. In other cases,where the goal may be system identification, the determination of thedamped natural frequency may just be the first step. A second step isrequired for determination of the real component of each pole/zero. Thereal component of every root can be determined using a numericalsolution to a constrained nonlinear optimization problem. In that casethe independent variables are the real components of each of the polesidentified, and the cost function is the sum of the error squaredbetween the complex value of the FRF and the complex value of thesynthesized FRF for the candidate point in the parameter space. Theerror squared is evaluated at each frequency of a two sided FRF toenforce the additional constraint that an FRF of a physical system willhave a response over negative frequencies which is the complex conjugateof the response at the corresponding positive frequency with the sameabsolute value, It may be advantageous to impose a weighting functionthat decreases with frequency to avoid under-emphasizing low frequencyresonances that span only a few frequency lines in the discrete FRF. Aweighting function that is constant over the lowest frequency range andthen decreases as an inverse function of frequency has a similar effectof having generated an FRF whose density is constant when the frequencyis plotted on a log scale. The amplitude of a first order low passfilter is a good approximation of such a function.

The constrained optimization problem is formulated by constructing atransfer function for a given combination of real components of roots.The transfer function is augmented by an additional real pole and anadditional real zero to account for general trends in the FRF that donot appear as salient resonances or anti-resonances in the FRFmeasurement. The number of augmentation poles and zeros can be extendedto include multiple real poles/zeros and multiple complex-conjugatepoles and zeroes. In the following formulation for the candidatetransfer function the known values are the ω_(p)(i), ω_(z)(i), nZeros,nPoles, and the unknowns are K, σ_(p)(i), σ_(z)(i), σ_(pa), and σ_(za).A constraint that all σ_(p) be less than zero is applied to ensure astable result. An additional constraint that all oz be less than zeromay optionally be applied to ensure a “minimum phase” result.

${\hat{H}(s)} = {K\frac{\begin{matrix}{\prod\limits_{\omega}^{nZeros}\;\left( {\left( {s + {{j\omega}_{z}(i)} - {\sigma_{z}(i)}} \right)*} \right.} \\\left. \left( {s - {{j\omega}_{z}(i)} - {\sigma_{z}(i)}} \right) \right)\end{matrix}}{\begin{matrix}{\prod\limits_{k = 1}^{nPoles}\;\left( {\left( {s + {{j\omega}_{p}(k)} - {\sigma_{p}(k)}} \right)*} \right.} \\\left. \left( {s - {{j\omega}_{p}(k)} - {\sigma_{p}(k)}} \right) \right)\end{matrix}}*\frac{\left( {s - \sigma_{za}} \right)}{\left( {s - \sigma_{Pa}} \right)}}$

The cost function for the constrained optimization is the sum of theweighted errors squared between the transfer function estimate and theFRF measurement where the transfer function estimate is evaluated ats=jw for each frequency in the two-sided FRF measurement. In cases wherethe measured FRF is one sided, it is a simple matter to make it twosided by concatenating the original FRF to the end of the original FRFconjugated and order-reversed. The cost function is shown below.V( σ_(p) , σ_(z) )=Σ[W(w _(i))*(∥H({tilde over (ω)}_(i))−Ĥ(j{tilde over(ω)} _(i))∥²+∥(H({tilde over (ω)}_(i)))*−Ĥ(−j{tilde over (ω)} _(i))∥²)]

As mentioned one way to formulate the weighting function is as theabsolute Value of a first order low pass filter. The cutoff frequencyfor the filter determines at what frequency de-emphasizing the highfrequency data (to emulate log-frequency spacing) should begin. For auniformly spaced FRF, a reasonable value for the filter's cutofffrequency ω₀ can by 100 times the frequency spacing.

${W(\omega)} = {\frac{\omega_{0}}{{j\omega} + \omega_{0}}}$

The present invention is not intended to be limited to a system, device,or method which must satisfy one or more of any stated or implied objector feature of the invention and is not limited to the exemplaryembodiments described herein. Modifications and substitutions by one ofordinary skill in the art are considered to be within the scope of thepresent invention.

1. A method for determining the damped natural frequencies of a dynamicsystem comprising the actions of: producing a frequency responsefunction for the dynamic system; smoothing the frequency responsefunction; identifying a set of possible candidate frequency values bydetermining when a derivative of the frequency response functiontraverses through zero; sorting the set of possible candidate frequencyvalues based on amplitude; identify the next candidate in the sortedcandidate frequency values starting from the beginning of the set thathas not been eliminated; determining a range of influence by identifyinga lower root frequency and higher root frequency; eliminating thecandidate frequency value by comparing the candidate frequency valuewith a maximum or minimum amplitude in the range of influence and whenthe candidate frequency value is eliminated returning to the action ofidentify the next candidate; checking a direction of a change of phasein the range of influence occurs in an expected direction, and whendirection is opposite the expected direction, eliminating the candidatefrequency value and the returning to the action of identify the nextcandidate; determining if any frequency candidate values of lessersignificance are in the range of influence by identifying frequencycandidate values occurring later in the frequency candidate values setand when the frequency candidate value occurs later eliminating saidlater candidate frequency value and the returning to the action ofverifying the current candidate; determining if any frequency candidatevalues of greater significance are in the range of influence byidentifying frequency candidate values occurring earlier in the setfrequency candidate values and when the earlier frequency candidatevalue has not been eliminated and appears in the range of influence ofthe currently considered candidate, eliminate the currently consideredcandidate frequency and the returning to the action of verifying thenext candidate; eliminating the frequency candidate value by comparing anoise level in the frequency response function to the amplitude in theregion of a pole and when the candidate frequency value is eliminatedreturning to the action of identify the next candidate; and returning tothe action of identifying the next candidate until all frequencycandidate values have been identified.
 2. The method of claim 1, whereinthe action of determining a range of influence of a pole involvesdetermining the lower root frequency by finding the highest frequencypassing through a factor k (0<k<1) times the amplitude at the candidatefrequency with a positive slope.
 3. The method of claim 1, wherein theaction of determining a range of influence of a zero involvesdetermining the lower root frequency by finding the highest frequencypassing through a factor k (k>1) times the amplitude at the candidatefrequency with a negative slope.
 4. The method of claim 1, wherein theaction of determining a range of influence of a pole involvesdetermining the higher root frequency by finding the lowest frequencypassing through a factor k (0<k<1) times the amplitude at the candidatefrequency with a negative slope.
 5. The method of claim 1, wherein theaction of determining a range of influence of a zero involvesdetermining the higher root frequency by finding the lowest frequencypassing through a factor k (k>1) times the amplitude at the candidatefrequency with a positive slope.
 6. The method of claim 1, wherein theaction of checking further involves fitting a straight line and anintercept to a phase determined for the region of influence.
 7. Themethod of claim 1, wherein the action of checking further involvesunwrapping operation of locating instantaneous phase changes.
 8. Themethod of claim 1, wherein the action of smoothing involves using anaveraging window wherein a width of the averaging window increaseslinearly with frequency.
 9. The method of claim 1, wherein the sortedcandidate frequency values are stored in an array.
 10. A method fordetermining the damped natural frequencies associated with the poles ofa dynamic system comprising the actions of: producing a frequencyresponse function for the dynamic system; smoothing the frequencyresponse function; identifying a set of possible candidate frequencyvalues by determining when a derivative of the frequency responsefunction traverses through zero with negative slope; sorting the set ofpossible candidate frequency values based on amplitude; identify thenext candidate in the sorted candidate frequency values starting fromthe beginning of the set that has not been eliminated; determining arange of influence by identifying a lower root frequency and higher rootfrequency; eliminating the candidate frequency value by comparing thecandidate frequency value with a maximum amplitude in the range ofinfluence and when the candidate frequency value is eliminated returningto the action of identify the next candidate; checking a direction of achange of phase in the range of influence is negative and when directionis positive eliminating the candidate frequency value and the returningto the action of identify the next candidate; determining if thefrequency candidate values is of lesser significance are in the range ofinfluence by identifying frequency candidate values occurring later inthe frequency candidate values and when the frequency candidate valueoccurs later and is a frequency within the range of influence of apresently considered candidate, eliminating the later candidatefrequency value and the returning to the action of verifying a currentcandidate; determining if the frequency candidate values is of greatersignificance are in the range of influence by identifying frequencycandidate values occurring earlier in the frequency candidate values andwhen the frequency candidate value appear earlier eliminating thecurrent candidate frequency value and the returning to the action ofidentify the next candidate; eliminating the frequency candidate valueby comparing a noise level in the frequency response function to theamplitude in the region of a zero and when the candidate frequency valueis eliminated returning to the action of identify the next candidate;and returning to the action of identifying the next candidate until allfrequency candidate values have been identified.
 11. The method of claim10, wherein the action of determining a range of influence involvesdetermining the lower root frequency by finding the highest frequencypassing through a factor k times the amplitude at the candidatefrequency with a negative slope.
 12. The method of claim 10, wherein theaction of determining a range of influence involves determining thehigher root frequency by finding the lowest frequency passing through afactor k times the amplitude at the candidate frequency with a positiveslope.
 13. The method of claim 10, wherein the action of checkingfurther involves fitting a straight line and an intercept to a phasedetermined for the region of influence.
 14. The method of claim 10,wherein the action of checking further involves unwrapping operation oflocating instantaneous phase changes.
 15. A system for determining thedamped natural frequencies of a dynamic system comprising: a module forproducing a frequency response function for the dynamic system; a modulefor smoothing the frequency response function; a module for identifyinga set of possible candidate frequency values by determining when aderivative of the frequency response function traverses through a valueof zero; a module for sorting the set of possible candidate frequencyvalues based on amplitude; a module for identifying the next candidatein the sorted candidate frequency values starting from the beginning ofthe set that has not been eliminated; a module for determining a rangeof influence by identifying a lower root frequency and higher rootfrequency; a module for eliminating the candidate frequency value bycomparing the candidate frequency value with a maximum or minimumamplitude in the range of influence when the candidate frequency valueis eliminated returning to the module for identify the next candidate; amodule for checking a direction of a change of phase in the range ofinfluence occurs in an expected direction, and when direction isopposite the expected direction, eliminating the candidate frequencyvalue and the returning to the action of identify the next candidate; amodule for determining if any frequency candidate values of lessersignificance are in the range of influence by identifying frequencycandidate values occurring later in the frequency candidate values setand when the frequency candidate value occurs later and is a frequencywithin the range of influence of the currently considered candidate,eliminating said later candidate frequency value and the returning tothe action of verifying the current candidate; a module for determiningif any frequency candidate values of greater significance are in therange of influence by identifying frequency candidate values occurringearlier in the set frequency candidate values and when the earlierfrequency candidate value has not been eliminated and appears in therange of influence of the currently considered candidate, eliminate thecurrently considered candidate frequency and the returning to the actionof verifying the next candidate; a module for determining if thefrequency is in the range of influence of the frequency responsefunction prior to the module for smoothing; a module for eliminating thefrequency candidate value by comparing a noise level in the frequencyresponse function to the amplitude in the region of a pole and when thecandidate frequency value is eliminated returning to the action ofidentify the next candidate; and a module for returning to the action ofidentifying the next candidate until all frequency candidate values havebeen identified.
 16. The system of claim 15, wherein the module fordetermining a range of influence of a pole involves determining thelower root frequency by finding the highest frequency below thecandidate frequency passing through a factor k (k<1) times the amplitudeat the candidate frequency with a positive slope, and determining therange of influence of a zero involves determining a lower root frequencyby finding a highest frequency below the candidate frequency passingthrough a factor k (k>1) times the amplitude at the candidate frequencywith a negative slope.
 17. The system of claim 15, wherein the modulefor determining a range of influence of a pole involves determining thehigher root frequency by finding the lowest frequency greater than thecandidate passing through a factor k (k<1) times the amplitude at thecandidate frequency with a negative slope, and determining the range ofinfluence of a zero involves determining a lower root frequency byfinding a lowest frequency greater than the candidate passing through afactor k (k<1) times the amplitude at the candidate frequency with apositive slope.
 18. The system of claim 15, wherein the module forchecking further involves fitting a straight line and an intercept to aphase determined for the region of influence.
 19. The system of claim15, wherein the module for checking further involves unwrappingoperation of locating instantaneous phase changes.
 20. The system ofclaim 15, wherein the module for smoothing involves using an averagingwindow wherein a width of the averaging window increases linearly withfrequency.